Can graph neural network infer knot invariants?

The intriguing fact of physical knots is that their knot topology, not their specific shape, dictates their physical behavior. This phenomenon can be commonly found in many physical knots, such as entanglements of DNA and polymer chains and those in climbing ropes and shoelaces. It is therefore crucial to identify the topological properties of knots. In this talk, we use a data-driven approach to predict the topologically invariant properties of knots from their centerline conformation. By simulating the random diffusion of strings, we generate multiple groups of knots associated with different knot topologies. Each group contains knots of distinct shapes yet topologically equivalent conformations. Using a graph neural network, we first perform a classification task on the groups of knots and achieve high classification accuracy. Then, we train the neural net to predict the ropelength (which is a scalar-valued knot invariant) of knots and find that the neural network is capable of making the inference of the ropelength of knots which the neural net has not previously seen. Finally, we extend our approach to multi-filament knots, demonstrating that the data-driven approach may be generalized to characterizing physical systems consisting of multiple filaments.